The integral in your image is:

$\int (3x^4 + 2x^2 - x + 3) \, dx$

To solve this, we will integrate each term separately using basic power rules for integration.

1. Integrating $3x^4$: $\int 3x^4 \, dx = \frac{3x^5}{5}$

2. Integrating $2x^2$: $\int 2x^2 \, dx = \frac{2x^3}{3}$

3. Integrating $-x$: $\int -x \, dx = \frac{-x^2}{2}$

4. Integrating $3$: $\int 3 \, dx = 3x$

Now, combining all the results:

$\frac{3x^5}{5} + \frac{2x^3}{3} - \frac{x^2}{2} + 3x + C$

where $C$ is the constant of integration.

Would you like further explanation on any part of this process?

Here are some related questions:

1. How do you apply the power rule for integration?
2. What is the constant of integration and why is it important?
3. How do you integrate rational functions?
4. What are some common types of integrals you can solve?
5. How do you handle definite integrals?

Tip: When integrating polynomials, always remember to increase the exponent by one and divide by the new exponent!